337 - 11-2 - The Magnetic Dipole: d

Preliminaries: Taylor-expansion of the -function in the integrand of the definition of the -component of the vector potential at , , due to the -component of a localized current-distribution yields the multipole-expansion of the vector-potential, in which there are magnetic monopoles, dipoles, quadrupoles, etc,

The continuity-equation for a current-density which is always localized is,

We consider the dipole moment from ,

We can simplify by generalizing , and making the following two mathematical identities, where we need from CM 04 - 181 - de 14 - double levi civita formula derivation,

Adding and yields the statement ; the matrix-elements can then be rewritten as,

Thus, the magnetic-dipole-approximation to the vector-potential is found by combining and ,

Lets calculate the dipole magnetic field from ,